Travaux sur les symÃ©tries de Lie des Ã©quations aux ... - DMA - Ens

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34and let us seek necessary and sufficient conditions in order that there existsolutions (X, Y ) to the first auxiliary system defined precisely by:(3.38){ □0xx = Π 0 0,0, □ 0 xy l 1= Π 0 0,l 1, □ 0 y l 1y l 2= Π 0 l 1 ,l 2,□ j xx = Π j 0,0, □ j xy l 1 = Πj 0,l 1, □ j y l 1y l 2 = Πj l 1 ,l 2.3.39. Compatibility conditions for the first auxiliary system. As in Section2, the compatibility conditions for this system will simply be obtainedby computing the cross differentiations. The following statement generalizesLemma 2.31 and also provides a proof of it, in the case m = 1.Lemma 3.40. For all j, l 1 , l 2 , l 3 = 0, 1, . . ., m, we have the cross differentiationrelations(3.41)( ) ( ) m∑m∑□ j − □ j = − □ k y l 1y l 2y l 3 y l 1y l 3y l y l 1y l 2· □ j + □ k2y l 3y k y l 1y l 3· □ j .y l 2y kk=0Proof. To begin with, as a preliminary, let us generalize the Plücker identity(2.28). Let C 1 , C 2 , . . .,C m , D, E be (m + 2) column vectors in K m andintroduce the following notation for the m × (m + 2) matrix consisting ofthese vectors:(3.42) [C 1 |C 2 | · · · |C m |D|E].Extracting columns from this matrix, we shall construct m×m determinantswhich are modification of the following “fundamental” determinant(3.43) |C 1 | · · · |C m | ≡ ∣ ∣ ∣ ∣ C1 | · · · | j 1C j1 | · · · | j 2C j2 | · · · |C m∣ ∣ ∣ ∣ .Here and in the sequel, we use a double vertical line in the beginning and inthe end to denote a determinant. Also, we emphasize two distinct columns,the j 1 -th and the j 2 -th, where j 2 > j 1 , since we will modify them. Forinstance in this matrix, let us replace these two columns by the column Dand by the column E, which yields the determinant(3.44)∣ ∣ ∣ C1 | · · · | j 1D| · · · | j 2E| · · · |C m∣ ∣∣ ∣.In this notation, one should understand that only the j 1 -th and the j 2 -thcolumns are distinct from the columns of the fundamental m × m determinant(3.43). With this notation at hand, we can now formulate and prove apreliminary lemma that will be useful later.k=0
Lemma 3.45. The following quadratic identity between determinants holdstrue:(3.46)⎧∣ ∣ ∣ ∣ ∣ ⎪⎨ C1 | · · · | j 1D| · · · | j 2E| · · · |C n ∣ · ∣C1 | · · · | j 1C j1 | · · · | j 2C j2 | · · · |C n == ∣ ∣ ∣ ∣ ∣ C1 | · · · | j 1D| · · · | j 2C j2 | · · · |C n · ∣C1 | · · · | j 1C j1 | · · · | j 2E| · · · |C n ∣ −⎪⎩− ∣ ∣ ∣ ∣ C1 | · · · | j 1E| · · · | j 2C j 2| · · · |C n · ∣C1 | · · · | j 1C j1 | · · · | j 2D| · · · |C n .Proof. After some permutations of columns, this identity amounts to(3.47)⎧⎪⎨⎪⎩|C 1 | · · · |C m−2 |D|E | · |C 1 | · · · |C m−2 |C m−1 |C m | == |C 1 | · · · |C m−2 |D|C m | · |C 1 | · · · |C m−2 |C m−1 |E | −− ||C 1 | · · · |C m−2 |E|C m | · |C 1 | · · · |C m−2 |C m−1 |D |.To establish this identity, we introduce some notation. If A and B are verticalvectors in K m and if i 1 , i 2 = 1, . . .,m with i 1 denote(3.48) ∆ 2 i 1 ,i 2(A|B) :=∣ A ∣i 1B i1 ∣∣∣.A i2 B i2If |A 1 |A 2 |A 3 | · · · |A m | is a m×m determinant, and if i 1 , i 2 = 1, . . .,m withi 1 denote by M m−2i 1 ,i 2(A 3 | · · · |A m ) the (m−2)×(m−2) determinantobtained from the matrix [A 3 | · · · |A m ] by erasing the i 1 -th line and the i 2 -thline. Without proof, we recall an elementary classical formula(3.49)⎧⎨ |A 1 |A 2 |A 3 | · · · |A m | =⎩= ∑1i 1
Page 1 and 2: Joël M E R K E RÉcole Normale Sup
Page 3 and 4: §1. INTRODUCTIONSeveral physically
Page 5 and 6: e a collection of m analytic second
Page 7 and 8: 7where the indices j, l 1 vary in {
Page 9 and 10: This phenomenon could be explained
Page 11 and 12: This yields the prolongation of the
Page 13 and 14: X(x, y) and Y (x, y) such that it m
Page 15 and 16: 2.23. Compatibility conditions for
Page 17 and 18: This lemma is left to the reader; a
Page 19 and 20: simplification nor any reordering:(
Page 21 and 22: aleza, y por otra, las organizacion
Page 23 and 24: computational level (differential-g
Page 25 and 26: For instance, in the case m = 2, by
Page 27 and 28: in (3.11). The second equation that
Page 29 and 30: Similarly, the second equation take
Page 31 and 32: order derivatives of X and of the Y
Page 33: Lemma 3.32. The system yxx j = F j
Page 37 and 38: often be denoted by the sign “·
Page 39 and 40: 1Now, taking account of the factor
Page 41 and 42: conditions, totally equivalent to t
Page 43 and 44: 43remaining terms afterwards:(3.71)
Page 45 and 46: Here, the sign ≡ precisely means:
Page 47 and 48: Next, replacing plainly (3.64) in (
Page 49 and 50: 49the order of §3.73. We get:(3.89
Page 51 and 52: 51Multiplying by −2 and reorganiz
Page 53 and 54: ⎧Θ l 1y l 2 = −Ll 1l 1 ,l 1 ,y
Page 55 and 56: + 1 ∑4 δj l 1Hl k 2L k k,kk− 2
Page 57 and 58: + 1 ∑2 δj l 1Hl k 3M l2 ,kk+ 1 4
Page 59 and 60: In the hardest techical part of thi
Page 61 and 62: − 1 3+ 1 3− 1 4+ 1 4∑Hl k 1H
Page 63 and 64: − 1 3∑kL l 1l1 ,k,x Hk l 1+ 1 3
Page 65 and 66: Here, the sign ≡ means “modulo
Page 67 and 68: Thirdly, put j := l 2 in (3.108) wi
Page 69 and 70: 69correct. We get:(4.29)0 =?== −
Page 71 and 72: Next, apply the operator ∑ k Ll 2
Page 73 and 74: 73Writing term by term the substrac
Page 75 and 76: of the terms of the subgoal (4.29):
Page 77 and 78: 77+ 1 2∑kH k l 1 ,y l 2 Hl 2k15
Page 79 and 80: 79= − X xx Yx j + Y jm∑+++++l 1
Page 81 and 82: Our first task is to compute the de
Page 83 and 84: Replacing this expression of A k in
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85have the continuation(5.17) −y
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87and where thirdly (we are nearly
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89obtain:(5.23)III :=m∑m∑l 1 =1
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[GTW1989] GRISSOM, C.; THOMPSON, G.
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93Nonalgebraizable real analytic tu
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and of CR dimension m = n − d ≥
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coordinates z ′ = 2i ln(z/z p ),
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{x ∈ K n : |x| < ρ} for some ρ
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(2) A K-algebraic inversion mapping
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(1) The complex Lie algebra Hol(M,
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Proposition 3.1. Let t ↦→ G i (
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The first relation gives nothing, s
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with |β∗ k | ≥ 1 and integers
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Clearly, the left hand side is an a
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field X 1 ′ := h ∗ (X 1 ). Taki
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which yields after differentiating
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117Im w∆ n(ρ 4 )∆ n(ρ 1 )∆
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[∂ i,ei H ei (t ′ )] t ′ =He
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p = (w p , z p , ζ p , ξ p ) ∈
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Finite nondegeneracy is interesting
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such that |ω j i ∗ ,β∗ i(t,
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for k even, we have Γ k (z (k) ) =
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that for every local holomorphic se
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h(t) = ˜H(t, J l 0µ 0¯h(0)) with
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infinite families of pairwise non b
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functions w(z)). The coefficients R
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Then the Lie criterion states that
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y k ′ = λk k y k and w ′ = µ
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The last statements of Corollaries
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[Sha2000] SHAFIKOV, R.: Analytic co
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Here x = (x 1 , . . ., x n ) ∈ K
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Consequently, for the case κ = 2,
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are devoted to provide a general on
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for l = 1, . . .,m such that the lo
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Lemma 8.1. The following conditions
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155namely X ←→ X + X ←→ X .
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In terms of Sussmann’s approach [
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x κ−1 χ κ−1 + O(|x| κ ) + O
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Observe that these expressions are
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where the first term I involves onl
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I 5 = ∑ i 1I 6 = ∑ i 1 ,i 2I 7
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U 2 U κ−1 , we obtain the five f
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and U 2 U κ−1 , we obtain the fi
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following equations for j = 1, . .
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linear equations:(7.28) ⎧0 = Rx j
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175R j containing at least one part
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177[3] :∑σ∈S κ−1κδ l,....
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the values of the partial derivativ
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also appears some derivatives Q l
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[5] F. ENGEL; LIE, S.: Theorie der
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185Nonrigid sphericalreal analytic
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origin:{ ( ∣ ∣ )AJ 4 1∣∣∣
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I 2 characterizes equivalence to w
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y means of a fundamental, elementar
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2) When M is Levi nondegenerate at
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195holding in C { z ′ , z ′ , w
Page 197 and 198:
Fortunately for our present purpose
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We notice passim that S ≡ Q x (no
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for a certain local K-analytic new
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Conversely, if y xx (x) = F ( x, y(
Page 205 and 206:
205But we may also express the dual
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Then thanks to a straightforward ap
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and we then expand carefully the re
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explicit computation, so let us rew
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213+T ah3Q 2 a Q b∆(aa|b)∆(b|bb
Page 215 and 216:
215Vanishing Hachtroudi curvaturean
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to fix the ideas, it will be assume
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then by replacing the(so obtained)v
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and also in addition the fact that
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and this defines without ambiguity
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the leaves of this local foliation
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Then by differentiating with respec
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∂Here, the coefficients of the2 T
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231for (7.28):( )∂□ ·∂a □
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233Lie symmetriesof partial differe
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Differentiating the first equation
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Example 1.21. (Continued) With n =
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defined by the graphed equations:((
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for some two local analytic maps Π
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Proof. Let l = l ( x i , y j , yβ(
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complementary views on the same obj
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Classification problem 3.11. Classi
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Definition 4.10. L is an infinitesi
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Convention 5.2. The letters R will
Page 253 and 254:
Assuming that dimSYM(E 1 ) = 8, tak
Page 255 and 256:
Restarting from §4.1, let ϕ a Lie
Page 257 and 258:
Here, R j l 1 ,kare universal polyn
Page 259 and 260:
Lemma 6.34. Let M be a submanifold
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where the expressions D β,β1 are
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Then the sum L + L is tangent to M
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Lemma 7.11. ([CM1974, BER1999, Me20
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where(7.33) ∆ := ∑1kn∂ 2∂x
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with r, s being unspecified functio
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and moreover, all other, higher ord
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To conclude, we replace X xx so obt
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Definition 8.43. A real analytic hy
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Lemma 8.59. The function b depends
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where the letter r denotes an unspe
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(8.80) ⎧Y 1,1,1 = Y x 1 x 1 x 1 +
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(X 1 , X 2 , Y , Xy 1,X 2x, Y 2 x 1
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espect to x 1 and (8.96) 2 with res
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For the two unknowns Xyy 1 and X 2x
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Redeveloping the determinant, the v
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291are obtained by equating to zero
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Let (z 0 , c 0 ) = (x 0 , y 0 , a 0
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we deduce that Γ 5 is submersive (
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11.4. Regularity and jet parametriz
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Cramer’s rule, we get(11.13) ⎧
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in K[a, z] n+m and in K[x, c] p+m .
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of the cancellation properties (11.
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305II: Explicit prolongations of in
Page 307 and 308:
Define then the transformed jet ϕ
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Let λ ∈ N be an arbitrary intege
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311[Ol1986], [BK1989]):(Y(1.31)⎧
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+ [Y y 2 − 2 X xy ] y 1 y 2 + [
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for some nonnegative integers A, B,
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Once the correct theorem is formula
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By the classical formula for the de
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The induction formulas are⎧(3.4)
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We have to compute:( ) ⎛ ⎞∑
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Next, we gather the underlined term
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+ ∑k 1 ,k 2 ,k 3 ,k 4[−δ k 1,k
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Correspondingly, we identify the se
Page 331 and 332:
must be equal to the number of indi
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Y i1 ,i 2 ,i 3 ,i 4written in one o
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335conclusion, we have shown that (
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337Theorem 3.73. For every κ 1 an
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Theorem 3.79. For every integer κ
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+++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3
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343form:(4.12)κ+1Yκ j = Y jx κ +
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eing related to the number 2 in the
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Appying the chain rule, we may comp
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Secondly:(5.3)Y j i 1 ,i 2= Y jx i
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As explained before the statement o
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g j i 1 ,...,i λand similarly for
Page 355 and 356:
with respect to the variables x i 1
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Assuming F = F(x, y x ) to be indep
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359(III’) ⎧0 = ∑ (δ k 2)j 3H
Page 361 and 362:
Open problem 1.17. For n = 2 establ
Page 363 and 364:
⎧∣ ⎨X 1 xX 1+ y x 1 y x 1 ·1
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eplacing the third column by the se
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⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣
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e written under the specific form:(
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Indeed, the collection of equations
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3.12. Principal unknowns. As there
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375Sixthly:(3.22) ⎧δ k 1j 1Θ n+
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are a consequence of (I’), (I”)
Page 379 and 380:
379IV: BibliographyREFERENCES[Ar198
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[G1989] GARDNER, R.B.: The method o
Page 383 and 384:
[Me2005a] MERKER, J.: On the local
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